PHYSICAL REVIEW E 92, 022146 (2015)

Crossover behavior in interface depinning

Y. J. ChenLASSP, Physics Department, Cornell University, Ithaca, New York 14853-2501, USA

Stefano ZapperiCenter for Complexity and Biosystems, Department of Physics, University of Milano, via Celoria 26, 20133 Milano, Italy;

CNR–Consiglio Nazionale delle Ricerche, IENI, Via R. Cozzi 53, 20125, Milano, Italy;ISI Foundation, Via Alassio 11/c 10126 Torino, Italy;

and Department of Applied Physics, Aalto University, P.O. Box 14100, FIN-00076, Aalto, Finland

James P. SethnaLASSP, Physics Department, Cornell University, Ithaca, New York 14853-2501, USA

(Received 9 August 2014; revised manuscript received 29 July 2015; published 27 August 2015)

We study the crossover scaling behavior of the height-height correlation function in interface depinning inrandom media. We analyze experimental data from a fracture experiment and simulate an elastic line modelwith nonlinear couplings and disorder. Both exhibit a crossover between two different universality classes. Forthe experiment, we fit a functional form to the universal crossover scaling function. For the model, we vary thesystem size and the strength of the nonlinear term and describe the crossover between the two universality classeswith a multiparameter scaling function. Our method provides a general strategy to extract scaling properties indepinning systems exhibiting crossover phenomena.

DOI: 10.1103/PhysRevE.92.022146 PACS number(s): 64.60.av, 64.60.fd, 68.35.Ct, 05.40.−a

I. INTRODUCTION

Driven interfaces in random media display intriguingscaling laws that are common to a wide variety of phenomena,including fluid imbibition, crack front roughening, dislocationhardening, superconducting flux lines, the equilibrium motionof piles of rice down an incline, and domain wall motion inmagnets [1,2]. The scaling laws are commonly associated withan underlying depinning critical point that has been elucidatedby simple models for interface dynamics. These models havebeen extensively studied using continuum simulations [3–6],cellular automata [3,7–11], and field-theoretic ε expansions[3,12–17], providing a sophisticated picture of the nonequilib-rium phase transition and of the different universality classes.

The interface morphology is usually characterized bythe roughness exponent ζ , resulting from a coarse grainingoperation of the interface height function h(x). Namely, whenwe change all length scales by a factor b, or x → bx, thenstatistically h → bζ h – hence

h(x) ∼ b−ζ h(bx). (1)

For many experiments and simulations, it is convenient tomeasure ζ by computing the height-height correlation of theinterface

C(r) = 〈[h(x + r) − h(x)]2〉 ∼ r2ζ . (2)

[In Sec. II [18] we shall study a system with anomalousscaling, where the power law exhibited by C(r) differs fromthe universal rescaling exponent ζ . Rather than rescaling h

in such systems, one studies the rescaling of the correlationfunction directly, C(r) ∼ b−2ζ C(br) ∼ r2ζ . These systemsare multiaffine [18]: different moments of h will scale withdifferent exponents]. Here ζ should be uniquely determinedby which universality class the system belongs to. However,in practice, the observed ζ varies (see Table I) even for thesame type of system, such as paper wetting. Measuring a

single exponent for these systems may prove inadequate dueto the presence of crossover behavior between universalityclasses. This is a common source of confusion and controversy.If the crossover is gradual, an experiment or simulationmay measure an effective exponent ζeff intermediate betweenexisting theories and appear to demand a new theoreticalexplanation (i.e., universality class).

In Sec. II we analyze a straightforward experimentalexample of a crossover between two forms of rough-ness in two-dimensional fracture. There we introduce theuniversal crossover scaling functions and provide a briefrenormalization-group rationale.

In the remainder of the paper, we examine a more complextheoretical model. Crossovers, long studied in ordinary criticalphenomena, have now been studied for several interfacemodels [22]; however, theoretical studies have proven chal-lenging in different ways [26]. For thin film magnets, theexperiments [27–29] observe a crossover between short-rangeand mean-field universality classes as long-range dipolar fieldsare introduced, which can be done by changing the thicknessof the film. However, for models of that type, simulationsare challenging, because of both the long-range fields andthe striking zig-zag morphologies that emerge and competewith the avalanche behavior. Crossovers involving the tran-sition between depinning and sliding dynamics incorporatingperiodically correlated disorder [30] have also been studied.It is not typical, however, to study and report the universalscaling functions for these crossovers, a challenge we nowshall address.

We shall analyze a numerically tractable, but analyticallytricky, crossover [26]: the transition between the linear,super-rough, quenched Edwards-Wilkinson model (qEW) andthe nonlinear quenched KPZ model (qKPZ) [2,22]. In bothexperiment and theory, we focus on the crossover behavior ofthe height-height correlation function.

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Y. J. CHEN, STEFANO ZAPPERI, AND JAMES P. SETHNA PHYSICAL REVIEW E 92, 022146 (2015)

TABLE I. Roughness exponents in experiments. Table repro-duced from Ref. [1]. Notice that there is a wide range of ζ reported,even for the same experimental system.

Experiment ζ Reference

Fluid flow 0.73 [19]0.81 [20]

0.65–0.91 [21]Paper wetting 0.63 [22]

0.62–0.78 [23]Bacteria growth 0.78 [24]Burning fronts 0.71 [25]

II. CROSSOVER IN FRACTURE SURFACECORRELATIONS

Just as the critical exponent ζ is universal (independentof microscopic details, within a class of physical system), sotoo is the crossover behavior between universality classes. As asimple example, Santucci et al. [18] have measured a relativelysharp crossover between two regimes for two-dimensionalfracture (inset in Fig. 1). Because the fracture is done slowly,we can view the crack front as self-organizing to the depinningtransition for the crack front. Well below a critical distance r∗,they observe a power law C(r) ∼ r2ζ− with an exponent that

FIG. 1. (Color online) Crossover scaling in fracture roughness[18]. The inset shows experimental data for the height-heightcorrelation function C(r) = 〈[h(x + r) − h(x)]2〉 of a 2D fracturefront, generated by pulling apart two pieces of PMMA that have beensand-blasted and sintered together [18]. The three curves differ inthe size of the sand-grain beads; the relation between the bead sizeand the toughness fluctuations in the PMMA were not measured.The dashed lines show two different power-law critical regimes, withC(r) ∼ r2ζ− and C(r) ∼ r2ζ+ , governing the short- and long-distancescaling behavior: the crossover between these regimes is evident.Our fit gives ζ− = 0.63 and ζ+ = 0.32, within the experimentalists’suggested range ζ− = 0.6 ± 0.05 and ζ+ = 0.35 ± 0.05. The mainfigure shows a scaling plot of r−2ζ−C(r) versus r , with the curvesshifted vertically and horizontally to best collapse. The thick blackcurve is a one-parameter fit of the universal scaling function to thefunctional form in Eq. (4).

was interpreted as originating from coalescing cracks [31]or with Larkin scaling [32]. Well above r∗ they observe adifferent power law C(r) ∼ r2ζ+ consistent with the depinningtransition of a line [32–34]. The crossover between thesetwo universal power-law regimes should be described by auniversal crossover function [35], Cfrac:

C(r) ≈ C∗r−2ζ−Cfrac(r/r∗) (3)

independent of microscopic details. At small argumentsCfrac(X) must go to a constant, and at large argumentsCfrac(X) ∼ X2(ζ−−ζ+), so as to interpolate between the twopower laws. When analyzing different systems governed bythe same universal crossover, one may plot all the crossoversin a scaling plot, dividing the distances r on the ordinate bya system-dependent factor r∗ for each curve, and dividingthe magnitudes of the correlations on the abscissa by asystem-dependent constant C∗ (see Fig. 1). The resultingdata curves then should align, giving the universal functionCfrac(r/r∗).

To continue with this simple test case, we may fit theuniversal scaling function to an approximate functional form.(Indeed, we find it convenient to do a joint fit of the functionalform, the exponents, and the constants r∗ and C∗). To theextent that a guessed functional form reproduces the universalone, it is equivalent: advanced field-theoretic methods forcalculating exact scaling functions are not needed to analyzefuture experiments. However, judicious choices of functionalforms with the correct limiting behavior can greatly facilitatethis process. The interpolation 1/(1 + X−2(ζ−−ζ+)) has thecorrect limits, but its rather gradual crossover does not explainthe data. We may heuristically add a parameter n which atlarge values produces an abrupt crossover:

Cfrac(X) = [1 + (X2(ζ−−ζ+))−n]−1/n. (4)

This yields an excellent fit to the data with n ≈ 4 (see Fig. 1).Why is the scaling form of Eq. (3) expected? Briefly, the

renormalization group studies the behavior of systems undercoarse graining: describing the properties of a system at lengthscales changed by a factor b. One gets universal power lawswhen the system becomes invariant under repeated coarsegrainings: if C(r) → b2ζ C(r/b), under coarse graining by afactor b, then by coarse graining n times such that r = bn

one has C(r) ∝ b2ζn = r2ζ . In the case of a crossover, a fixedpoint is unstable to some direction λ in system space. Then asmall initial λ grows under rescaling by some factor b1/φ , soC(r,λ) → b2ζ−C(r/b,λb1/φ). Now rescaling until bn = r , wehave

C(r,λ) → b2nζ−C(r/bn,λbn/φ) = r2ζ−C(1,λr1/φ)

= r2ζ−Cfrac(λφr), (5)

where we choose Cfrac(X) = C(1,X1/φ). If the unstabledirection flows to a new fixed point with a different ζ+,that behavior will be reflected in the large-X dependenceC(X) ∼ X2(ζ−−ζ+) [36, Sec. 4.2]. Note that different physicalsystems will have different overall scales of height fluctuations,so we must have an overall scale C∗ for each experiment. (If theexperiments fall into a parameterized family, C∗ will dependsmoothly on the parameters, giving analytic correctionsto scaling as discussed in Sec. IV). Note, though, that the

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rescaling factor r∗ for lengths, while it will still vary fromone system to another, now depends on λ with a power-lawsingularity, as r∗ = 1/λφ ; within the renormalization group,λ measures how far along the unstable direction the originalsystem was poised. In particular, r∗ becomes large as λ → 0,as in that limit the unstable fixed point remains in control.

The three experiments depicted in Fig. 1 started withdifferent bead sizes. If all other features of the experimentare held fixed, one may assume that the control parameterλ depends in some smooth way on the bead size. Had weseveral values of bead size, we could then extract values forthe universal crossover exponent φ.

In the following sections, we shall perform a far moresophisticated version of this type of analysis. By exhaustivelyvarying system size and nonlinearity in an interface growthmodel, we shall not only generate universal two-variablefunctional forms for the correlation crossover scaling function,but will be able to make predictions about both the dependenceof the crossover length scale (corresponding to r∗) and thedependence of the correlation amplitude (corresponding toC∗) on the control parameters. A rich, nuanced understandingof the model behavior thus emerges.

III. LINE DEPINNING MODEL

The equations of an interface in a disordered environmentmay be written generally as follows. Let the one-dimensionalinterface, h(x,t) be driven by a force H (t) through a disorderedenvironment with a local quenched random force η(h(x),x):

∂h

∂t= γ∇2h + λ(∇h)2 + η(h(x),x) − k〈h〉x + H (t). (6)

Here γ is a surface tension, and λ is the coefficient ofthe KPZ term. The KPZ term controls lateral spreadingof the interface, breaks the statistical tilt symmetry, andchanges the universality class [1]. H (t) is a slowly increasingexternal driving force. Our simulations are done with a latticeautomaton; the lattice naively might be thought to break thisstatistical tilt symmetry, but simulations have long shown thatthe model faithfully describes both universality classes [37].

The term −k〈h〉x is borrowed from simulations of magnets,where it represents the demagnetization force [38], approxi-mating the effects of the long-range dipolar field cost of a netadvance in the front. This restoring force “self-organizes” thedepinning transition to the fixed point, allowing simulationsto access many metastable states, without having to enforcean actual quasistatic field. It is known [39] that this restoringforce does not produce loop corrections to the renormalizationgroup equations and therefore does not change the universalityclass of the problem. We have confirmed numerically that itseffects are small for our crossover and appear irrelevant. Asthe restoring force makes the simulation vastly more efficient,we include this restoring force, but we do not include k in ourscaling analysis.

IV. ANALYSIS OF CROSSOVER SCALING

Using the automaton simulation employed in Ref. [40], wetune λ/γ from 0 to 5 and observe how the resulting behaviorchanges. Figure 2 shows how the front morphology qualita-tively changes while we increase the nonlinear parameter λ.Notice that with increasing λ the fronts between events areflatter than at small λ.

According to Eq. (2), naively one would assume we couldrecover the exponent ζ by defining an effective exponent ζeff

to be half the local-log slope of the height-height correlationfunctions (Fig. 3). From other numerical studies, for qEW,we expect ζEW 1.25. (Cellular automata [8,41] modelsshow ζEW = 1.25 ± 0.01; continuous string models [42] foundζEW ≈ 1.26.) For qKPZ, we expect ζKPZ = 0.63 [11,43].However, there are two things about Fig. 3 worth noting:(1) the slope measure of ζ drifts between 0.63 and 1.0 aswe change λ, and (2) the measured value is never greater thanone as is naively expected for the linear qEW model. Thedropoff at r ∼ L/4 is due to the periodic boundary conditions.

The second issue has a known resolution: for ζ > 1, whenthe interface is “super-rough,” the height cannot grow fasterthan linearly with distance, so the height-height correlationfunction cannot directly exhibit a power law larger than one[44]. This so-called anomalous scaling [45–47] implies that the

FIG. 2. (Color online) Crossover of qKPZ to qEW model. Fronts generated from 128 × 256 simulations with the nonlinear KPZ termcoefficients set to (a) λ = 0, (b) λ = 0.001, (c) λ = 0.1, (d) λ = 5. The random colors (shades) represent the area between each pinned front.The morphology of the interfaces changes dramatically as λ increases.

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FIG. 3. (Color online) Local log slope. The measured local-logslope, d log C/d log r of the height-height correlation function forvarying λ and k = 0.01. The lower dashed red line is ζKPZ = 0.63 asexpected for the KPZ universality class. The upper dashed blue line isζSR = 1.0, the largest growth allowed for super-rough interfaces. Thethin lines show the predictions of our fits (Sec. IV), and the dashedblack line is the fit value of ζKPZ. The sharp cutoff in the curves at larger is due to the periodic boundary condition, which forces ζeff = 0 atr = L/2; the larger L curves have later cutoffs. As expected, for smallλ the curves are described by the EW behavior ζSR = 1 at small r ,while for large λ they are well described by ζKPZ. For intermediatevalues of λ ∼ 0.1, we observe a clear transition from EW behaviorat small r to KPZ behavior at intermediate r , before being cut off bythe finite size effects.

exponent ζ is reflected not in the distance dependence of thecorrelation function, but rather in its system-size dependence.We thus consider the finite-size scaling form

CEW(r|L) ∼ L2ζEW (r/L)2CEW(r/L); (7)

the roughness exponent ζEW may be estimated by thesystem-size dependence of the magnitude of CEW. Note thatthe periodic boundary conditions implies that CEW(r|L) =CEW(−r|L) = CEW(L − r|L); near r = L/2 the correlationfunction reaches a peak (and ζeff vanishes, as in Fig. 3). ThusX2CEW(X) = (1 − X)2CEW(1 − X). To control the sharpnessof the peak in the correlation function at X = 1/2, in analogyto the crossover sharpness parameter n of Eq. (4), we introducenEW giving a transition between the two power laws:

X2CEW(X) = {(X2)−nEW + [(1 − X)2]−nEW}−1/nEW . (8)

For qKPZ [Fig. 2(c)], the correlation function in a system sizeL takes the finite-size scaling form

CKPZ(r|L) = Ar2ζKPZCKPZ(r/L), (9)

where we introduce nKPZ to form a periodic functional form

X2ζKPZCKPZ(X) = {(X2ζKPZ )−nKPZ

+ [(1 − X)2ζKPZ ]−nKPZ}−1/nKPZ . (10)

The drift in the exponent ζ , however, demands a study of thescaling near the unstable qEW fixed point and the functionalform of the resulting crossover scaling function. The role ofλ in generating the crossover from qEW to qKPZ has onlybeen studied qualitatively [4,7,10,11], with no full descriptionof the crossover scaling [26]. The crossover describes the RGflow from the qEW fixed point to the qKPZ as the relevantparameter λ is added. The scaling form for the height-heightcorrelation function is thus that of a relevant variable λ addedto the qEW scaling:

C(r|L,λ) = L2ζEWC(r/L,λφr). (11)

For λ � 0, C(r|L,λ) → CKPZ(r|L); therefore,

C(r/L,λφr)

→ A(λ)r2ζKPZCKPZ(r/L)/L2ζEW

= A(λ)(r/L)2ζKPZL−2(ζEW−ζKPZ)CKPZ(r/L)

= Aanalytic(λ)(r/L)2ζKPZ (λφL)−2(ζEW−ζKPZ)CKPZ(r/L). (12)

FIG. 4. (Color online) Height-height correlation function. Thenumerics generated with an automata code (symbols) are welldescribed by Eq. (16) (black curves) with fit parameters φ =1.0 ± 0.4, ζKPZ = 0.65 ± 0.04, ζEW = 1.1 ± 0.15, nCross = 1.0 ±0.7, B = 2.5 ± 6.0. nEW = 0.27 ± 0.03, nKPZ = 1.1 ± 0.6, A0 =2.7 ± 7, A1 = 770 ± 900, and A∞ = 0.3 ± 1. (A fit constrained tothe expected values of ζEW and ζKPZ give slightly worse, but acceptablefits, with the other parameter estimates within the quoted ranges). Thelegends denote L and λ for each simulated correlation function; allruns had k = 0.01. The errors quoted are a rough measure of thesystematic error [48], as described in the text, and are representativeof the differences we find using different weightings and functionalforms. (They are much larger than the statistical errors). Theamplitude dependence is captured by the scaling form. Three of the 12parameters (ζEW, ζKPZ, and φ) are universal critical exponents, three(A0, A1, and A∞) describe the nonuniversal dependence of an overallheight scale on parameters, two describe finite-size effects, and onlytwo (nCross and B) are needed to describe the universal crossoverfunction to the accuracy shown.

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FIG. 5. (Color online) Rescaled spectral function S(q|L,λ) forthe height fluctuations. Rescaling by the naive RG power q1+2ζEW

does collapse the data for vanishing λ (up to a significant noise):the anomalous scaling for the super-rough interface in real space isnot manifested in Fourier space. The theory curves are given by theFourier transform of the same best fit shown in previous figures.

Here A(λ) is in general a nonuniversal prefactor for theKPZ correlation function. As with C∗ in Sec. II, A isexpected to vary [49] with the parameters of the problem.

Here it has a typical smooth variation Aanalytic(λ), times asingular piece Asingular(λ) due to the EW fixed point: A(λ) =Aanalytic(λ)Asingular(λ). We derive the power-law divergence ofthe amplitude

Asingular(λ) = λ−2(ζEW−ζKPZ)φ (13)

by noting that C(r/L,λφr) must be a scaling function with onlyinvariant combinations of r , L, λ.

We must also have C(r/L,0) ∼ (r/L)2CEW(r|L). Usingthese limits, Eq. (14), and the fact that A(λ) gets large asλ → 0 and small as λ → ∞, we can construct a function thatcrosses over between these two limits:

C(r|L,λ) = Aanalytic{[Asingular(λ)BCKPZ(r/L)]−nCross

+CEW(r/L)−nCross}−1/nCross

= L2ζKPZC(X,Y ), (14)

where X = r/L and Y = λφr , and CKPZ and CEW aredefined in Eqs. (7)–(10). We expand the analytic functionAanalytic = (A0 − A∞)/(1 + A1λ) + A∞ in a form analytic atzero and saturating at large λ at A∞[50], and we include arelative scale factor B. Finally, we vary the sharpness of thecrossover with nCross, just as we did in the experimental studyof fracture [Eq. (4)].

The theoretical curves in were fit to the data in Figs. 3, 4,and 5, deleting the noisy half near r = L/2 in the first, andusing weights σ 2 ∼ √

r/L designed to equalize the emphasison each decade. The errors quoted are a rough estimate of thesystematic error [48] given by quadratically exploring fits withroughly twice the ξ 2 of the best fit.

(a) (b)

FIG. 6. (Color online) Scaling collapse of the height-height correlation function. The correlation-function data of Fig. 4 are collapsedto illustrate the crossover between EW and KPZ-dominated lengths (r < r∗ and r > r∗, respectively; see also Refs. [28,29]). Here r∗ =L[B(Lλφ)2(ζEW−ζKPZ)]−1/(2−2ζKPZ) is the distance where the EW and KPZ components of the correlation function are equal in magnitude, andC∗ = λ−2φ(ζEW−ζKPZ)r2ζKPZ factors out the dependence expected in the KPZ regime (hence yielding flat behavior for r � r∗). (b) Effects ofanalytic corrections to scaling are also factored out; all of the curves lie on a universal curve apart from the effects of the finite system sizes(causing each curve to drop on the right). (a) The analytic correction to scaling has a significant impact on the scaling collapse: ignoring itin the analysis would produce significant errors in critical exponents and scaling functions. The thick blue curve shows the scaling functionprediction for the crossover [which, if x = r/r∗, can be shown to be C/C∗ = Bx2(x2nCross + x2nCrossζKPZ )−1/nCross ].

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Note that this gives us a universal function of three variables(r , L, and λ). Note that it predicts a singularity at small λ inthe form of a divergent amplitude Asing ∼ λ−2(ζEW−ζKPZ)φ in theqKPZ correlation function [Eq. (9)] as λ → 0. This universalsingularity in the amplitude (corresponding to a prediction ofC∗ in Sec. II) explains the amplitude dependence seen in Fig. 4.There is an analogous universal amplitude dependence seen forthe Heisenberg → Ising crossover at small Ising anisotropy[36, Sec. 4.1].

We can use the scaling form of the correlation functionC(r|L,λ) to derive other, less complex crossover scalingfunctions traditionally studied in interface depinning problems[30,42]. The spectral scaling function is equal to the Fouriertransform of our correlation function (with subtleties at q = 0):

S(q|L,λ) = |h(q)|2 =∫

r

C(r|L,λ) exp(iqr) dr

∼ q−1−2ζEWS(λ−φq,qL). (15)

Here the universal spectral crossover scaling function S(X,Y )can be written in terms of our universal correlation crossoverscaling function [C(X,Y ) from Eq. (11)] as

S(X,Y ) =∫

dz exp(iz)Y 2ζEWC(z/Y ,z/X) (16)

depicted in Fig. 5. Here it is known that S does not haveanomalous scaling; the power laws ζEW and ζKPZ can be readoff from the slopes at large and small Y . More ordinary,single-variable scaling functions can be derived from our two-variable scaling functions, such as that governing the averagewidth of an interface [given by the zero Fourier componentof S(q) or an integral of C(r)]; such scaling functions [asfor the fracture scaling function of Eq. (3)] allow traditionalscaling collapses (as in Fig. 1). However, one should notethat our analyzed simulations extend to λ ∼ 1, where analyticcorrections to scaling, as vividly illustrated in Fig. 6(a), wouldlikely invisibly distort the resulting scaling collapses. It is anadvantage of multivariable scaling fits that they both allowthe incorporation of such analytic corrections (extending therange of applicability) and force their incorporation (exposingweaknesses of the naive theory).

V. CONCLUSIONS

In this paper, we have analyzed the scaling properties foran experimental 2D fracture front and a model of an interfacemoving in random media, focusing on the crossover scalingof the roughness. The experimental system is successfullymodeled using a one-variable universal scaling function withone free parameter, controlling the sharpness of the transition.The theoretical model, the crossover from the qEW to theqKPZ universality class with the addition of a nonlinear term,allows us to estimate the complete universal scaling functionfor the height-height correlation function including both finite-

size effects and the nonlinear effects of the tuning parameterλ, while satisfying known limits given by the renormalizationgroup.

We emphasize the importance of our sophisticated useof the scaling forms and corrections predicted from therenormalization group. Figure 1 illustrates that not only thepower laws, but the entire functional form of the crossover,is a universal property that should be reported. Equation(4) is an effective one-parameter way of embodying thesharpness of the crossover, which we use also in the theoreticalanalysis of Sec. IV for both the crossover and the effectsof periodic boundary conditions. Figures 3, 4, and 5 showhow different experimental characterizations of the roughnessof an interface can be simultaneously fit with a singlefunctional form. Figure 6(a) vividly indicates the importanceof analytic corrections to scaling in extending the validityof the theoretical analysis to smaller systems and fartheroutside the critical region. Only systems with λ < 10−3 willfollow the scaling behavior without incorporating the analyticcorrections, while the entire crossover is faithfully representedin Fig. 6(b) by including them in the fit.

By developing functional forms for the correlation func-tions [40], we gain the flexibility of incorporating analyticcorrections, multiple scaling variables, and a systematic erroranalysis while allowing the quantitative reporting of theuniversal scaling functions. One should note that the parameterestimation errors quoted here are large compared to moretraditional scaling analyses. In part this is due to our estimationof the relevant systematic errors [48]; statistical errors wouldbe perhaps an order of magnitude smaller. In part, however, thisis due to our incorporation of known but usually ignored con-founding factors; analytic corrections to scaling and crossovereffects will invisibly distort the results of more direct mea-surements, and the drift in exponents quoted in the literaturein critical phenomena often exceeds the error estimates.

It is challenging but satisfying to develop these functionalforms. Measuring and fitting them is far more physicallyintuitive and less technically demanding than direct field-theoretic calculations [42] (although theoretical calculationsoften form important inspiration for choosing functional forms[40]). Successful functions are parsimonious in the number ofadjustable parameters, and developing them often forces oneto develop a far more complete understanding of the physicsof the system under consideration.

ACKNOWLEDGMENTS

This work is supported by NSF and CNR through MaterialsWorld Network: Cooperative Activity in Materials Researchbetween US Investigators and their Counterparts Abroadin Italy (NSF DMR 1312160) and through NSF PHY11-25914. S.Z. acknowledges support from the Academy ofFinland FiDiPro program, project 13282993 and the EuropeanResearch Council through the Advanced Grant No. 291002SIZEFFECTS.

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become important at long distances from the critical point. Forexample [36, Section 3.3], λ is presumably not the naturalmeasure of the unstable direction in parameter space; in generalthat would be an analytic function uλ ∝ λ + O(λ2) of λ andother parameters.

[50] Indeed, simulations at λ = 2 and λ = 5 indicate a saturationof the amplitude of the KPZ power-law scaling. This suggeststhat Aanalytic(λ) ∝ 1/Asingular(λ) at larger λ. In the range weconsider, an asymptotically flat asymptote suffices to capture thisvariation; independent fits of the amplitude at each λ includingthe larger systems give comparable fits and other parameterswithin our ranges.

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